Puzzle for July 1, 2022 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
Add F to both sides of eq.5: E – F + F = D + F + F which becomes eq.5a) E = D + 2×F In eq.6, replace E with D + 2×F (from eq.5a): B + D = D + 2×F + F – D which becomes B + D = 3×F Subtract D from each side of the equation above: B + D – D = 3×F – D which becomes eq.6a) B = 3×F – D
Hint #2
In eq.3, replace B with 3×F – D (from eq.6a), and E with D + 2×F (from eq.5a): C + D = 3×F – D + D + 2×F which becomes C + D = 5×F Subtract D from each side of the above equation: C + D – D = 5×F – D which becomes eq.3a) C = 5×F – D
Hint #3
In eq.2, substitute 3×F – D for B (from eq.6a): D = A + 3×F – D In the above equation, subtract 3×F from both sides, and add D to both sides: D – 3×F + D = A + 3×F – D – 3×F + D which becomes eq.2a) 2×D – 3×F = A
Hint #4
Substitute 2×D – 3×F for A (from eq.2a), 3×F – D for B (from eq.6a), and 5×F – D for C (from eq.3a) in eq.4: 2×D – 3×F + F = 3×F – D + 5×F – D which becomes 2×D – 2×F = 8×F – 2×D Add 2×F and 2×D to both sides of the equation above: 2×D – 2×F + 2×F + 2×D = 8×F – 2×D + 2×F + 2×D which becomes 4×D = 10×F Divide both sides by 4: 4×D ÷ 4 = 10×F ÷ 4 which makes D = 2½×F
Hint #5
Substitute (2½×F) for D in eq.2a: 2×(2½×F) – 3×F = A which becomes 5×F – 3×F = A which makes 2×F = A
Hint #6
Substitute 2½×F for D in eq.6a: B = 3×F – 2½×F which makes B = ½×F
Hint #7
Substitute 2½×F for D in eq.3a: C = 5×F – 2½×F which becomes C = 2½×F
Hint #8
Substitute 2½×F for D in eq.5a: E = 2½×F + 2×F which makes E = 4½×F
Solution
Substitute 2×F for A, ½×F for B, 2½×F for C and D, and 4½×F for E in eq.1: 2×F + ½×F + 2½×F + 2½×F + 4½×F + F = 26 which simplifies to 13×F = 26 Divide both sides of the above equation by 13: 13×F ÷ 13 = 26 ÷ 13 which means F = 2 making A = 2×F = 2 × 2 = 4 B = ½×F = ½ × 2 = 1 C = D = 2½×F = 2½ × 2 = 5 E = 4½×F = 4½ × 2 = 9 and ABCDEF = 415592